On 24 Mar 2015, at 02:58, Bruce Kellett wrote:
meekerdb wrote:
On 3/23/2015 5:11 PM, Bruce Kellett wrote:
LizR wrote:
On 24 March 2015 at 08:02, meekerdb <[email protected] <mailto:[email protected]
>> wrote:
That every number has a unique successor for one.
"Let's call the first number that doesn't have a unique successor
n..."
Can you prove that n+1 exists and is unique? Just asserting it is
not sufficient.
Of course you can prove it from Peano's axioms. But that's the
point: that what you can prove is relative to the axioms you assume
(and the rules of inference too). So when someone asserts some
propositions and says see everyone believes these, they are very
intuitive, and then he proceeds to prove something implausible
which counter-intuitive then one is justified in wondering whether
the axioms apply to the real world.
Right, that is what I meant. Proof is only within an axiom system
Yes. It is equivalent with belief. Different machines, different
beliefs. Proving is a relative notion, it has no universal definition,
unlike computability. But classical proof by enough rich systems obeys
all the same modal logic of self-reference G and G* (by a theorem of
Solovay).
-- the hard problem is demonstrating that those axioms are the ones
relevant to the world we experience.
Yes. Now mathematical logic+computer science study both the math of
the 3p actions of beliefs (brain activity, proof theory) and the set
of models capable of satisfying the beliefs. At the propositional
level they are captured by modal operator (in the ideal case of simple
correct machine). []p = true in all models, and <>p = there is a modal
satisfying p. There is a "Galois adjunction" between theories and
models, like between equations and sets of solutions.
There are brains, minds, and *plausibly* a reality, whose nature is to
decide.
Bruno
Bruce
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